Question

What is the meaning of the limit of a function?

Answer 1

The statement `lim_(x→a)f(x) =L` means: as `x` gets closer to `a`, `f(x)` gets closer to `L`.

Explanation:

The precise definition is:

For any real number `ε>0`, there exists another real number `δ>0` such that if `0<|x-a|<δ`, then `|f(x)-L|<ε`.

Consider the function `f(x) =(x^2-1)/(x-1)`.

If we plot the graph, it looks like this:

We can't say what the value is at `x=1`, but it does look as if `f(x)` approaches `2` as `x` approaches `1`.

Let's try to show that `lim_(x→1) (x^2-1)/(x-1) = 2`.

The question is, how do we get from `0<|x-1|<δ` to `|(x^2-1)/(x-1)-2| <ε`?

We must start with some value of `ε` and then find a find a corresponding value for `δ`.

Let's start with

`|(x^2-1)/(x-1) -2| =|((x+1)(x-1))/(x-1)-2| = |x+1-2| = |x-1|<ε`

The other condition is

`|x-1| < δ`

The definition fits exactly if `δ = ε`.

We have just shown that for any `ε`, there is a `δ` so that `|f(x)−2|<ε` when `0<|x−1|<δ`.

So we have shown that

`lim_(x→1) (x^2-1)/(x-1) = 2`